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Question
The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is
Options
- \[\frac{\pi^4}{2}\]
- \[\frac{\pi^4}{4}\]
0
none of these
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Solution
0
\[\int_{- \pi}^\pi \sin^3 x \cos^2 x d x\]
\[ = \int_{- \pi}^\pi \sin x\left( 1 - \cos^2 x \right) \cos^2 x dx\]
\[Let\ \cos x = t, then - \sin x dx = dt, \]
\[When\, x = - \pi, t = - 1, x = \pi, t = - 1\]
\[\text{Therefore the integral becomes}\]
\[ \int_{- 1}^{- 1} - \left( 1 - t^2 \right) t^2 dt\]
\[ = 0\]
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